Real and complex behaviour of the base change function (was: The "cheta" function)

[bo198214]

Simply pick all the points whose real part is equal to log((2k+1)*pi), k a non-negative integer, and whose imaginary part is equal to (2m+1)/2*pi, for m an integer.

Exponentiating once will get you to +/- (2k+1)*pi*i, which exponentiating again will get you to -1.

Thats good for visualization.

Note that this grid of points covers the entire right half of the complex plane, so that when we iteratively perform logarithms, we can always find points close to the real line.

But still I dont get this. Iterated (branches of) logarithms take any point to one of the fixed points of exp. Why do they come arbitrary close to the real axis with increasing n?
I hope the picture will clarify.

Even if this is the case, and I trust you enough to believe it, the bigger question is whether the singularities come arbitrarily close to *any* point of the real axis where the base change is defined. Or are there points with a certain neighborhood that does not contain a singularity of any \( f_n \)? If so, we would just use such a point for the powerseries development.

because we have to wind in and around singularities to make sure we are in the primary branch of the logarithm of base eta.

I think we anyway have to specify a cut system, as we have (branching) singularities. The functions \( f_n \) are multivalued (if we continue passing a cut).
If I understand that right, you set the cuts of \( f_n \) to be the points that would be mapped to the negative real axis of any involved \( \log_\eta \).
See, if you define the value via a path, then you will usually get different values for non-homotopic paths (i.e. any deformation of one into the other would cross a singularity).
So this is the same as a multivalued function. (The situation gets even worse if singularities appear on certain but not all branches.)
The proper path that leads to using primary branch of the logarithm would then be a path that does not cross any cut of \( f_n \).
I think the cut system is like a labyrinth and hope you can provide a picture of it.